Statistics 2
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BUSINESS MATHEMATICS
AND
STATISTICS - 523
Q.1
(a)
Ans.
TIME SERIES:
SECOND
ASSIGNMENT
What is a time series? Why time series analysis is of interest to the business?
A “time series” consists of numerical data collected, observed or recorded at regular intervals of time each hour, day, month, quarter or year. More specifically, it is any set of data in which observations are arranged in a chronological order. Examples of time series, the weekly prices of wheat in Lahore, the monthly consumption of electricity in a certain town, the monthly total of passengers carried by rail, the quarterly sales of a certain fertilizers, the annual rainfall at Karachi for a number of years, the enrolment of students in a college or university over a number of years and so forth. The observations in a time series, denoted by y1, y2.....,
yt, are usually made at equidistant points of time or they are associated with equal intervals of time (t). In order to examine a time series, the first step is to plot the given series on a graph, taking time intervals (t) along the X axis, as the independent variable, and the observed values (yt) on the Yaxis, as the dependent variable. Such a graph will show various types of fluctuations and other points of interest. INTEREST TO THE BUSINESS Time series is very useful in business to observe the outcomes of distinct causes of variation. For example the following table shows the number of bags (in hundreds) of fertilizer sold by a certain dealer.
Ejaz Alam Khan - H 5279752
# 1
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Year 1995 1996 1997 1998
I 72 79 94 125
SECOND
Quarters II III 98 79 122 101 141 128 143 135
ASSIGNMENT
IV 106 143 160 187
Now in the following lines the data as a time series is plotted and comment made.
Ejaz Alam Khan - H 5279752
# 2
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
The above historigram obtained by plotting the given time
Ejaz Alam Khan - H 5279752
# 3
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
series, shows that the sales have risen for the second quarter, have fallen for the third quarter and then have risen to a higher point for the fourth quarter every year. This graph also reveals that the sales, on the whole, have risen over 4 years. The graph further suggests that by smooting out the irregularities, the annual rate at which the sales have increased, may be ascertained.
Ejaz Alam Khan - H 5279752
# 4
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
COMPONENTS OF A TIME SERIES: A typical time series may be regarded as composed of four basic types: •
Secular Trend this is a long term movement that persists for many years and indicates the general direction of the change of observed values. In other words, it refers to the smooth, broad, steady and regular movement of a time series in the same direction, showing a gradual rise or fall within the date. In the following graph, the long term trend of sales is shown which measures the trend and helps in ascertaining the rate of change to be used for further estimates. It also heps to business planning and in studying the other variations:
Ejaz Alam Khan - H 5279752
# 5
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 6
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 7
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
•
Seasonal Variations. The seasonal variations which are mainly caused by the change in seasons, are short term movements occurring in a periodic manner. In the following graph the fluctuations are repeated with more or less the same intensity within a specific period of one year or shorter based on causes for seational variation such as weather conditions. etc.
•
Cyclical Fluctuations. These are the long period oscillations about the long term trend, which tend to occur in a more or less regular pattern over a period of certain number of years.
•
Irregular or Random Variations. These are irregular and unsystematic in nature. They occur in a completely unpredictable manner as they are caused by some sporadic or unusual events such as floods,
Ejaz Alam Khan - H 5279752
# 8
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
droughts, strikes, fires, etc.
Ejaz Alam Khan - H 5279752
# 9
BUSINESS MATHEMATICS
Q.1
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Sales of electronic coffee makers at a discount department store over the last five years are shown below. Using weights of 0.1, 0.2, 0.3 and 0.4 prepare a forecast for next year.
Ans. Year
Sales
Weights
Forecast
Percentage
0.0
1992
9,000
1993 1994 1995 1996
7,000 8,000 8,500 8,200
0.1 0.2 0.3 0.4
700 1,600 2,550 3,280
10% 20% 30% 40%
Total
40,700
1.0
8,130
100%
Base year
Ejaz Alam Khan - H 5279752
# 10
BUSINESS MATHEMATICS
Q.2
AND
STATISTICS - 523
(a)
SECOND
ASSIGNMENT
If U = {1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4} B = {3,5,6} Then show A ↔ Β by Venn diagram.
Ans.
A ↔ Β = { 3 }
Ejaz Alam Khan - H 5279752
# 11
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 12
BUSINESS MATHEMATICS
Q.2
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Solve the following system of equations:
Ans.
Ejaz Alam Khan - H 5279752
# 13
BUSINESS MATHEMATICS Q. 3
AND
STATISTICS - 523
SECOND
ASSIGNMENT
A manufacturer makes two types of products, I and II, at two plants x and y. In the manufacture of these products the following pollutants result. Sulphur dioxide, carbon monoxide, and particulate matter. At either plant, the daily pollutants resulting from the production of product I are: 300 pounds of sulphur dioxide 100 pounds of carbon monoxide 200 pounds of particulate matter and of product II are: 400 pounds of sulphur dioxide 50 pounds of carbon monoxide 300 pounds of particulate matter To satisfy the federal regulations, these pollutants are removed. Suppose that the daily cost in rupees for removing each pound of pollutant at plant x is: 5 rupees/pound of sulphur dioxide 3 rupees/pounds of carbon monoxide 2 rupees/pounds of particulate matter and at plant y is: 8 rupees/pound of sulphur dioxide 4 rupees/pounds of carbon monoxide 1 rupees/pounds of particulate matter Using matrices, calculate the daily cost of removing all pollutants from one of the plants in the manufacture of one of the products.
Ans. Cost of removal of Product I at Plant X =Rs. 2200 Cost of removal of Product I at Plant Y =Rs. 3000 Cost of removal of Product II at Plant X =Rs. 2750 Cost of removal of Product II at Plant Y =Rs. 3700
Ejaz Alam Khan - H 5279752
# 14
BUSINESS MATHEMATICS
Q. 4
(a)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Suppose a manufacturer finds that the cost (in rupees) making x units of a product is given by: If the rate of production is maintained at a constant units per day, find the rate at which the cost is changing with time when the level of production is x = 36 units.
Ans.
By derivative
Ejaz Alam Khan - H 5279752
# 15
BUSINESS MATHEMATICS
Q.4
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Suppose a manufacturer is limited by the firm’s production facilities to a daily output of at the most 80 units and suppose that the daily cost and revenue functions are (rupees) (rupees) Assuming that all units produced are sold, how many units should be manufactured daily to maximize the profit?
Ans.
Maximum profit is obtained = Marginal Revenue = Marginal Cost
Ejaz Alam Khan - H 5279752
# 16
BUSINESS MATHEMATICS
Q.5
(a)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
A manufacturer determines that the marginal cost in rupees is given by: Find the cost function C(x), assuming that the fixed cost (cost when x = 0 units are produced) is Rs. 30.
Ans.
By antiderivative
Ejaz Alam Khan - H 5279752
# 17
BUSINESS MATHEMATICS
Q.5
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
In the beginning of 1984 (t=0) zinc was being consumed at a rate of 578,850 short tons per year and the consumption rate was increasing at 4.5% per year. If we assume an exponential growth model for the rate of consumption, then Estimate the total amount of zinc that will be used from 1984 to 1994.
Ans.
Ejaz Alam Khan - H 5279752
# 18
AND
STATISTICS - 523
Q.1
(a)
Ans.
TIME SERIES:
SECOND
ASSIGNMENT
What is a time series? Why time series analysis is of interest to the business?
A “time series” consists of numerical data collected, observed or recorded at regular intervals of time each hour, day, month, quarter or year. More specifically, it is any set of data in which observations are arranged in a chronological order. Examples of time series, the weekly prices of wheat in Lahore, the monthly consumption of electricity in a certain town, the monthly total of passengers carried by rail, the quarterly sales of a certain fertilizers, the annual rainfall at Karachi for a number of years, the enrolment of students in a college or university over a number of years and so forth. The observations in a time series, denoted by y1, y2.....,
yt, are usually made at equidistant points of time or they are associated with equal intervals of time (t). In order to examine a time series, the first step is to plot the given series on a graph, taking time intervals (t) along the X axis, as the independent variable, and the observed values (yt) on the Yaxis, as the dependent variable. Such a graph will show various types of fluctuations and other points of interest. INTEREST TO THE BUSINESS Time series is very useful in business to observe the outcomes of distinct causes of variation. For example the following table shows the number of bags (in hundreds) of fertilizer sold by a certain dealer.
Ejaz Alam Khan - H 5279752
# 1
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Year 1995 1996 1997 1998
I 72 79 94 125
SECOND
Quarters II III 98 79 122 101 141 128 143 135
ASSIGNMENT
IV 106 143 160 187
Now in the following lines the data as a time series is plotted and comment made.
Ejaz Alam Khan - H 5279752
# 2
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
The above historigram obtained by plotting the given time
Ejaz Alam Khan - H 5279752
# 3
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
series, shows that the sales have risen for the second quarter, have fallen for the third quarter and then have risen to a higher point for the fourth quarter every year. This graph also reveals that the sales, on the whole, have risen over 4 years. The graph further suggests that by smooting out the irregularities, the annual rate at which the sales have increased, may be ascertained.
Ejaz Alam Khan - H 5279752
# 4
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
COMPONENTS OF A TIME SERIES: A typical time series may be regarded as composed of four basic types: •
Secular Trend this is a long term movement that persists for many years and indicates the general direction of the change of observed values. In other words, it refers to the smooth, broad, steady and regular movement of a time series in the same direction, showing a gradual rise or fall within the date. In the following graph, the long term trend of sales is shown which measures the trend and helps in ascertaining the rate of change to be used for further estimates. It also heps to business planning and in studying the other variations:
Ejaz Alam Khan - H 5279752
# 5
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 6
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 7
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
•
Seasonal Variations. The seasonal variations which are mainly caused by the change in seasons, are short term movements occurring in a periodic manner. In the following graph the fluctuations are repeated with more or less the same intensity within a specific period of one year or shorter based on causes for seational variation such as weather conditions. etc.
•
Cyclical Fluctuations. These are the long period oscillations about the long term trend, which tend to occur in a more or less regular pattern over a period of certain number of years.
•
Irregular or Random Variations. These are irregular and unsystematic in nature. They occur in a completely unpredictable manner as they are caused by some sporadic or unusual events such as floods,
Ejaz Alam Khan - H 5279752
# 8
BUSINESS MATHEMATICS
AND
STATISTICS - 523
SECOND
ASSIGNMENT
droughts, strikes, fires, etc.
Ejaz Alam Khan - H 5279752
# 9
BUSINESS MATHEMATICS
Q.1
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Sales of electronic coffee makers at a discount department store over the last five years are shown below. Using weights of 0.1, 0.2, 0.3 and 0.4 prepare a forecast for next year.
Ans. Year
Sales
Weights
Forecast
Percentage
0.0
1992
9,000
1993 1994 1995 1996
7,000 8,000 8,500 8,200
0.1 0.2 0.3 0.4
700 1,600 2,550 3,280
10% 20% 30% 40%
Total
40,700
1.0
8,130
100%
Base year
Ejaz Alam Khan - H 5279752
# 10
BUSINESS MATHEMATICS
Q.2
AND
STATISTICS - 523
(a)
SECOND
ASSIGNMENT
If U = {1,2,3,4,5,6,7,8,9,10} A = {1,2,3,4} B = {3,5,6} Then show A ↔ Β by Venn diagram.
Ans.
A ↔ Β = { 3 }
Ejaz Alam Khan - H 5279752
# 11
BUSINESS MATHEMATICS
AND
STATISTICS - 523
Ejaz Alam Khan - H 5279752
SECOND
ASSIGNMENT
# 12
BUSINESS MATHEMATICS
Q.2
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Solve the following system of equations:
Ans.
Ejaz Alam Khan - H 5279752
# 13
BUSINESS MATHEMATICS Q. 3
AND
STATISTICS - 523
SECOND
ASSIGNMENT
A manufacturer makes two types of products, I and II, at two plants x and y. In the manufacture of these products the following pollutants result. Sulphur dioxide, carbon monoxide, and particulate matter. At either plant, the daily pollutants resulting from the production of product I are: 300 pounds of sulphur dioxide 100 pounds of carbon monoxide 200 pounds of particulate matter and of product II are: 400 pounds of sulphur dioxide 50 pounds of carbon monoxide 300 pounds of particulate matter To satisfy the federal regulations, these pollutants are removed. Suppose that the daily cost in rupees for removing each pound of pollutant at plant x is: 5 rupees/pound of sulphur dioxide 3 rupees/pounds of carbon monoxide 2 rupees/pounds of particulate matter and at plant y is: 8 rupees/pound of sulphur dioxide 4 rupees/pounds of carbon monoxide 1 rupees/pounds of particulate matter Using matrices, calculate the daily cost of removing all pollutants from one of the plants in the manufacture of one of the products.
Ans. Cost of removal of Product I at Plant X =Rs. 2200 Cost of removal of Product I at Plant Y =Rs. 3000 Cost of removal of Product II at Plant X =Rs. 2750 Cost of removal of Product II at Plant Y =Rs. 3700
Ejaz Alam Khan - H 5279752
# 14
BUSINESS MATHEMATICS
Q. 4
(a)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Suppose a manufacturer finds that the cost (in rupees) making x units of a product is given by: If the rate of production is maintained at a constant units per day, find the rate at which the cost is changing with time when the level of production is x = 36 units.
Ans.
By derivative
Ejaz Alam Khan - H 5279752
# 15
BUSINESS MATHEMATICS
Q.4
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
Suppose a manufacturer is limited by the firm’s production facilities to a daily output of at the most 80 units and suppose that the daily cost and revenue functions are (rupees) (rupees) Assuming that all units produced are sold, how many units should be manufactured daily to maximize the profit?
Ans.
Maximum profit is obtained = Marginal Revenue = Marginal Cost
Ejaz Alam Khan - H 5279752
# 16
BUSINESS MATHEMATICS
Q.5
(a)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
A manufacturer determines that the marginal cost in rupees is given by: Find the cost function C(x), assuming that the fixed cost (cost when x = 0 units are produced) is Rs. 30.
Ans.
By antiderivative
Ejaz Alam Khan - H 5279752
# 17
BUSINESS MATHEMATICS
Q.5
(b)
AND
STATISTICS - 523
SECOND
ASSIGNMENT
In the beginning of 1984 (t=0) zinc was being consumed at a rate of 578,850 short tons per year and the consumption rate was increasing at 4.5% per year. If we assume an exponential growth model for the rate of consumption, then Estimate the total amount of zinc that will be used from 1984 to 1994.
Ans.
Ejaz Alam Khan - H 5279752
# 18
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